Thermo-mechanical large deformation response and ...pamies.web.engr.illinois.edu/Publications_files/IJP06.pdf · Thermo-mechanical large deformation response and constitutive modeling

International Journal of Plasticity 22 (2006) 581–601

www.elsevier.com/locate/ijplas

Thermo-mechanical large deformation responseand constitutive modeling of viscoelasticpolymers over a wide range of strain rates

and temperatures

Akhtar S. Khan *, Oscar Lopez-Pamies 1, Rehan Kazmi

Department of Mechanical Engineering, University of Maryland Baltimore County, Baltimore, MD 21250, USA

Received 14 April 2005Available online 28 November 2005

Abstract

A phenomenological one-dimensional constitutive model, characterizing the complex and highlynonlinear finite thermo-mechanical behavior of viscoelastic polymers, is developed in this investiga-tion. This simple differential form model is based on a combination of linear and nonlinear springswith dashpots, incorporating typical polymeric behavior such as shear thinning, thermal softening athigher temperatures and nonlinear dependence on deformation and loading rate. Another model, ofintegral form, namely the modified superposition principle (MSP), is also modified further and usedto show the advantage of the newly developed model over MSP. The material parameters for bothmodels are determined for Adiprene-L100, a polyurethane based rubber. The constants once deter-mined are then utilized to predict the behavior under strain rate jump compression, multiple stepstress relaxation loading experiment and free end torsion experiments. The new constitutive modelshows very good agreement with the experimental data for Adiprene-L100 for the various finite load-ing paths considered here and provides a flexible framework for a three-dimensional generalization.� 2005 Published by Elsevier Ltd.

Keywords: Polymers; High strain rate; Modeling; Temperature dependence; Viscoelasticity; Finite deformation

0749-6419/$ - see front matter � 2005 Published by Elsevier Ltd.

doi:10.1016/j.ijplas.2005.08.001

* Corresponding author. Tel.: +1 410 455 3301; fax: +1 410 455 1052.E-mail address: [email protected] (A.S. Khan).

1 Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia,PA 19104-6315, USA.

mailto:[email protected]

582 A.S. Khan et al. / International Journal of Plasticity 22 (2006) 581–601

1. Introduction

Polymers are proven to have an enormous and intriguing range of desirable features.This is basically due to the linking and arrangement of their smaller units or monomers.The synthesis, arrangement and conformations of these monomeric units result in a classof materials with a range of varied properties (Painter and Coleman, 1997). The manyattractive mechanical characteristics of polymers have made it desirable to choose thesematerials over traditional materials for numerous types of applications such as binder-constituent in explosives, load-bearing components, jet engines modules, etc. As the usesof polymers increase, a thorough understanding of the mechanical behavior of these mate-rials becomes vital in order to perform innovative and economical designs of variouscomponents.

Considerable progress has been made in developing mathematical models for the smallstrain regime under a specific narrow spectrum of strain rates and temperatures (see themonographs by Lockett, 1972; Christensen, 1982). Much less progress has been madefor multiaxial finite deformation behavior under a wide range of strain rates and temper-atures from a continuum point of view. Continuum approach ignores the molecular natureof matter and treats it in terms of laws of elasticity for solids and laws of fluid dynamicsand viscous flow for liquids (Ogden, 1997). Polymers, however, have more complicatedproperties as they display both elastic and viscous type response at different strain ratesand temperatures. Some literature though can be found for modeling large deformationsin polymers like Haupt et al. (2000); Ehlers and Markert (2003). Haupt et al. (2000) haveproposed a model for finite viscoelasticity. The model is based on a multiplicative split ofthe deformation gradient into a thermal and mechanical part. However, they did not applyto experimental observations, and thus validity of their model is not proven. Reese (2003)has also developed a material model for the thermo-viscoelastic behavior of rubber-likepolymers. It is based on transient network theory. In the case of micromechanics basedmodeling, a high degree of refinement has been reached to model finite deformation ofpolymers. However, these micromechanics based models are extremely difficult to useand rely on assumptions that have been questioned (e.g., Argon et al., 1995). Some ofthese micromechanical models can be found in the literature (Boyce et al., 1988; Bergstromand Boyce, 1998; Reese, 2003; Drozdov and Dorfmann, 2003; Makradi et al., 2005). Thereare also other studies that deal with polymer modeling based on over stress (Krempl andKhan, 2003; Colak, 2005). Some readers may find a paper by Liu et al. (2006) interesting,which is based on modeling of shape-memory polymers.

The main objective of this paper is to develop a simple and flexible phenomenologicalconstitutive model to characterize the observed time and temperature dependent mechan-ical response of soft polymers under finite deformation. The developed model is shown topredict the observed behavior of Adiprene-L100, a polyether urethane-based rubber,under various one-dimensional, time as well as temperature dependent loading experi-ments. Extensive uniaxial experimental data with the experimental details for this polymercan be found in Lopez-Pamies and Khan (2002). In that study, the dimensions of the uni-axial specimens were 1.00 in. (25.4 mm) in length and 0.80 in. (20.5 mm) in diameter. Hol-low tubular specimens were used to perform the free end torsion experiments. The innerdiameter of the specimens was approximately 0.55 in. (13.97 mm) and the outer diameterwas 0.675 in. (17.145 mm). The length of the specimen was approximately 0.85 in.(21.59 mm). All the quasi-static experiments were performed under displacement/angle

A.S. Khan et al. / International Journal of Plasticity 22 (2006) 581–601 583

control mode on a MTS axial/torsional servo-hydraulic 810 system. Dynamic experimentwas performed using the conventional compression split-Hopkinson pressure bar appara-tus. The specimen for the dynamic experiment was solid disk of Adiprene-L100 of 0.42 in.(10.67 mm) diameter and 0.13 in. (3.30 mm) thickness. The detailed description of thedynamic setup and SHPB technique can be found in previous papers (Liang and Khan,1999; Khan and Zhang, 2001).

Previous studies (Lopez-Pamies and Khan, 2002; Gray et al., 1997) have confirmed thatthe material response of Adiprene- L100 is almost totally viscoelastic, with very littleviscoplastic deformation. As a result, only viscoelastic modeling is considered in thispaper.

2. Classical linear theory of viscoelasticity

Substantial development has been attained in modeling the small strain regime behaviorof polymeric materials. The well-known classical linear theory of viscoelasticity (Flugge,1967; Christensen, 1982) describes fairly well the characteristic behavior of polymers suchas creep, relaxation, and loading-rate dependence in a qualitative manner. However, thequantitative description by using this linear viscoelasticity is generally limited to very nar-row loading-rate and temperature regimes.

The classical linear theory of viscoelasticity can basically be presented by two majorforms: hereditary integrals or differential forms. Even though the differential constitutiveforms are not as general (Lockett, 1972) as the hereditary integral representations, they aremore commonly used. This fact is mostly due to the more appealing usage of parameterssuch as stress/strain and stress/strain rates, as opposed to relaxation and creep kernels.Furthermore, the differential forms can be directly related to the intuitive spring and dash-pot diagrams. It is recalled here that a linear spring constitutive equation is of the formrs = Ees, where rs and es are the stress and the strain in the spring, respectively. E is theconstant modulus of the spring and implies a linear elastic stress–strain relationship. Sim-ilarly, a linear dashpot constitutive equation is of the form rd ¼ g_ed, where rd and _ed arethe stress and strain rate in the dashpot, respectively. g is the constant viscosity and impliesa Newtonian viscous effect.

Since the origin of the infinitesimal viscoelastic theory, numerous modifications havebeen introduced to improve on predictions of real material behavior from different com-binations of springs and dashpots (Lubarda et al., 2003; Bardenhagen et al., 1997). Weconsider now a popular combination of springs and dashpot, the so-called standard solidmodel, shown by Fig. 1 and composed of a Maxwell element (linear spring and dashpot inseries) with a linear spring in parallel. The stress in the standard solid is the sum of the

E1

η

E2

Fig. 1. Three-element standard solid model.


stresses in the Maxwell element and the linear spring. The governing equation for the sys-tem can be readily derived as

gE1

drdt

þ r ¼ gE1

ðE1 þ E2Þdedt

þ E2e. ð1Þ

Here g, E1, and E2 are material constants; r and e are the stress and strain, respectively.Note the structure of the constitutive equation (1). Only the current strain and stress,and current strain and stress rates are the arguments of the system. It can be shown thatthe standard solid model captures qualitatively the trends of general material behavior suchas creep, stress-relaxation, and loading-rate sensitivity, even for large strains. This point willbe elaborated further and utilized later to develop a new constitutive model. Eq. (1) is just aparticular equation for the specific case of the standard solid model. More generally, anylinear viscoelastic differential model can be expressed as follows (Christensen, 1982):

p0 þ p1o

otþ p2

o2

ot2þ � � �

� �rðtÞ ¼ q0 þ q1

o

otþ q2

o2

ot2þ � � �

� �eðtÞ; ð2Þ

where pi and qi are material constants. The differential form of Eq. (2) solely involves cur-rent strain, stress, strain rates, stress rates and higher derivatives. These arguments, as al-ready discussed, have a clear physical insight. In addition, the material parameters in Eq.(2) can be readily determined from simple experiments.

Some important observations should be made regarding Eq. (2). Firstly, the timedependent component of stress relaxation processes is described by a summation of expo-nential terms, namely

rðtÞ ¼ c0 þ c1e� t

s1 þ c2e� t

s2 þ � � � þ cne�tsn ; ð3Þ

where ci are constants and si are relaxation times (i = 1, . . .,n). It has been noted that thetime dependent component of the stress relaxation of polymers, say g(t), very often followsa simple power law-type behavior g(t) = tn (Lockett, 1972). In order for Eq. (3) to ade-quately approximate this feature, many exponential terms may be required. In essence, lin-ear viscoelastic differential constitutive relations necessitate a large number of materialparameters to approximate actual transient stress relaxation responses. Also, the powerlaw behavior of creep and relaxation can be represented using the concept of fractionalderivatives as shown by Lion (2001) and Lion and Kardelky (2004). Furthermore, poly-mers do not exhibit constant viscosity as assumed by the linear dashpot constitutive rela-tion rd ¼ g_ed. Instead, they exhibit shear thinning (or more rarely, shear thickening). Asthe name suggests, shear thinning implies that the viscosity decreases with increasing strainrate. This phenomenon has been attributed to the variation of the rate of polymer chain‘‘disentanglement’’ with increasing strain rate (Painter and Coleman, 1997). Also, underdifferent temperature conditions they are also known to exhibit totally different responses.At low temperatures they behave in a brittle elastic manner; at high temperatures, muchabove the glass transition the behavior is more �rubbery� (viscoelastic). There is a hugedrop in modulus when the temperature is increased above the glass transition temperature,indicating a shift in behavior from �glassy� to �rubbery� (Bardenhagen et al., 1997). Becauseof these peculiar mechanical properties of polymeric materials, the linear theory of visco-elasticity is unable to model closely the observed response and thus there is a need for anon-linear theory of viscoelasticity.


3. Nonlinear viscoelasticity

Many nonlinear theories of viscoelasticity have been proposed (Lockett, 1972; Chris-tensen, 1982). Most of them follow a similar pattern. Analogous to the linear case,they are based on the axiom that the stress in a viscoelastic material depends on theentire deformation history; or alternatively, the strain in a viscoelastic material dependson the entire stress history. The stress–strain constitutive models can be put in thethree major groups: the multiple integral, the single integral, and the differential rela-tions. Some of these relations have a physical foundation whereas others are purelyempirical.

The most general multiple integral constitutive relation for a nonlinear viscoelasticmaterial is given by the Green–Rivlin theory (Green and Rivlin, 1957). This constitutiverelation was developed around the dependence of the current stress (strain) state on theentire history of the deformation (stress), with the assumption of material isotropy. Underthese hypothesis and further mathematical constraints, Green and Rivlin developed a con-stitutive law based on an expansion of multiple integrals with multivariable relaxationfunctions as kernels. A convenient form is (Lockett, 1972):

RðtÞ ¼Z t

�1½I � /1 � T 1 þ /2 �M1� ds1 þ

Z Z�1

½I � /3 � T 1 � T 2 þ I � /4 � T 12

þ /5 � T 1 �M2/6 �M1 �M2� ds2 þZ Z Z

�1

½I � /7 � T 123 þ I � /8 � T 1 � T 23

þ /9 � T 1 � T 2 �M3 þ /10 � T 12 �M3 þ /11 � T 1 �M2 �M3

þ /12 �M1 �M2 �M3� ds3 þ � � � ; ð4Þ

where

dsx ¼ ds1 ds2 ds3 . . . dsx;

Mx ¼ _EðsxÞ; T x ¼ tr ðMxÞ;T xy ¼ tr ðMxMyÞ; T xyz ¼ tr ðMxMyMzÞ.

ð5Þ

For hereditary materials the material functions, or stress relaxation functions, /1 and/2 are functions of t � s1; /3, . . .,/6 are functions of t � s 1, t � s2; /7, . . .,/12 are functionsof t � s1, t � s2, and t � s3. The polynomial expression in Eq. (4) may be continued tohigher orders where the applicability becomes totally unpractical. Eq. (4) is one form ofthe Green–Rivlin Theory. The explicit expression in Eq. (4) involves many material func-tions and it has been shown that the determination of such functions is nearly impossibleas it requires unrealistic numbers of experiments (of the order of hundreds). In synthesis,the multiple integral constitutive relations introduced by the Green–Rivlin Theory arebased on very reasonable constraints. The behavior is assumed isotropic as well as themathematical assumptions involved with the development of the theory are not restrictive.These reasonable constraints introduce a high complexity in the model. The material func-tions involve in Eq. (4) depend on several integration variables. Also, the experimentaldetermination of these multivariable material functions requires an impractical numberof tests. The complexity of this model and the large amount of experimental data required


for the determination of the material parameters resulted in a very limited use. Despite thisfact, the Green–Rivlin theory provides an idealistic nonlinear viscoelastic theory fromwhich other models can be derived and compared.

Significant efforts have been made to reach more applicable models for nonlinear vis-coelastic materials than the Green–Rivlin theory (McGuirt and Lianis, 1970; Findley,1968; Pipkin and Rogers, 1968). Smart and Williams (1972) have compared three singleintegral models which are much easier to implement. A number of physical and semi-empirical single integral constitutive relations have also been proposed (Caruthers et al.,2004). These models are also mathematically very complicated to use. Here, one of thesesingle integral models, the modified superposition principle (MSP), is presented and uti-lized to model Adiprene-L100. The reason to consider this single integral form of consti-tutive model in particular is based on its simplicity and foundation. More specifically, theidea is to compare an existing semi-empirical stress–strain constitutive law to the phenom-enological differential constitutive model developed in this study.

3.1. The modified superposition principle

The MSP was first introduced by Leaderman in the early 1940�s (Smart and Williams,1972). After some experimental work on polymeric fibers, Leaderman concluded that thestrain behavior of the fibers under creep could be separated into time and stress dependentparts (Ward and Sweeney, 2004), that is

eðtÞ ¼ krþ hðrÞDðtÞ ð6Þwith an associated convolution integral

eðtÞ ¼ krðtÞ þZ t

0

Dðt � sÞ dhðrðsÞÞds

ds; ð7Þ

where e(t) and r(t) are the engineering strain and true stress, respectively; k is a materialconstant, h(r) and D(t) represent the separation of the stress and time dependent parts ofthe creep response. This was named the modified superposition principle by Lai and Find-ley (1968). Note that the true stress r(t), i.e., stress with respect to the deformed configu-ration is being used in Eq. (7). Leaderman also considered formulations in terms ofengineering stress, i.e., the stress with respect to the undeformed configuration, as wellas some other nonlinear stress measures (see Lockett, 1972 for further details). The mod-ified superposition principle can also be represented with the stress as a functional of strainhistory. Since it is a nonlinear theory, there are no analytical inversion formulae as in lin-ear viscoelasticity. The stress response of a nonlinear viscoelastic material under relaxa-tion, in a manner similar to Eq. (6), could be separated into a time dependent, D(t),and a strain dependent, h(e), part, such that

rðtÞ ¼ keþ hðeÞDðtÞ ð8Þwith the corresponding convolution integral

rðtÞ ¼ keðtÞ þZ t

0

Dðt � sÞ dhðeðsÞÞds

ds. ð9Þ

Eq. (9) represents the MSP-type constitutive relation utilized later to model the responseof Adiprene-L100.


3.2. Approach to the development of the modified standard solid model (MSSM)

A popular approach to model nonlinear finite viscoelastic behavior has been the use ofdifferential constitutive relations (Lubarda et al., 2003; Bardenhagen et al., 1997). Variousdifferential forms are based on spring and dashpots combinations, i.e., the foundations ofthese models are on linear viscoelastic constitutive relations. The standard solid model,Fig. 1, is one such type of model. The constitutive equation for this model has already beenshown by Eq. (1)

Bardenhagen et al. (1997) have also considered Eq. (1) as a starting point to model finitedeformation of polymers. However, their work was primarily focused on the generaliza-tion to three-dimensional case of both viscoelastic and viscoplastic deformation regimes.

In order to acquire more insight into the behavior of the standard solid model, it isimportant to consider numerical simulations for constant strain rate loading and stressrelaxation processes, as well as the corresponding analytical solutions to these loadings.The analytical solution for constant strain rate loading (i.e., _e ¼ _e0) is readily determinedfrom Eq. (1) yielding

r ¼ E2eþ g_e0 � g_e0e�E1e=g_e0 . ð10Þ

Note that the initial modulus for any given constant strain rate loading is simply the sumof the moduli of the two linear springs in the model. The asymptotic behavior for suffi-ciently large strains is linear, rð1Þ ¼ E2eþ g_e. Fig. 2(a) depicts the strain rate sensitivityof the standard solid model (see also Bardenhagen et al., 1997). From this figure as well asfrom Eq. (10) it is clearly concluded that the standard solid model under constant strainrate and finite deformation presents a bilinear behavior, that is, linear effective modulusand linear asymptotic strain rate hardening. This is a good initial approximation to theobserved behavior of typical polymeric materials under such a loading. Similarly, the ana-lytical solution for single-step stress relaxation processes (i.e., e = e0H(t); where H(t) is theHeaviside function) readily follows from Eq. (1)

rðtÞ ¼ E2e0 þ E1e0e�E1t=g. ð11Þ

Fig. 2(b) shows the schematic of stress-relaxation exhibited by using the standard solidmodel. Again, the trends of typical polymeric behavior are well captured by this simplemodel making it a good starting point for the development of an advanced model. How-ever, in order to adapt and exploit the differential form of the standard solid model tocharacterize finite nonlinear deformation of real polymers, various changes with regardsto the definitions of stress, stress rate, strain and strain rate need to be introduced for finitedeformation. The engineering strain e needs to be replaced by adequate and objective mea-sure for the finite deformation regime. The selected parameter chosen for strain should bebased on the deformed configuration. There exist several choices for deformation mea-sures, i.e., true strain etrue = ln (V), where V is the left Cauchy tensor, the LagrangeanAlmansi–Hemel strain (E), the Eulerian Almansi–Hemel (e) strain, etc. For this study, truestrain was selected as the most appropriate unit for the strain measure. Also, the true orCauchy stress is being utilized as the measure of the response of the material under finitedeformation.

In the determination of the constitutive equation for the modified standard solid model(MSSM), various springs and dashpots were utilized. Fig. 3 depicts the arrangement usedfor this study. It basically represents a Maxwell element connected in parallel with a

η 2

η 1

ε ε2>

2

ε1

1

2ε

t

σ

ε

1

. .

.

ε.ε.

ε.

ε.

ε.

σ

t

ε2 > ε1

E2ε2

E2ε1

( E1 +E2 )ε2

( E1 +E2 )ε1

ε

t

ε2

ε1

a

b

Fig. 2. (a) Strain rate sensitivity of the standard solid model. (b) Relaxation response of the standard solid model.

E1

η1

E2

η2

Fig. 3. Rheological schematic of the modified standard solid model.


Kelvin solid (except E2 is not a linear spring). The spring E1 is considered to be a materialconstant dependent only on operating temperature, whereas the other spring, E2, isassumed to be a nonlinear spring dependent on the level of deformation as well as temper-ature as a significant change in the modulus with temperature has been observed. Theequation for viscosity used for the two dashpots in the study was not taken to be a con-stant quantity, rather it was considered to be a function of strain rate and temperature.This is consistent with the term used by other researchers on polymers, as viscosity has


been known to depend on both, rate of loading as well as the temperature of the material.The original form of the viscosity term, which has been modified here, was takenfrom Bird et al. (1977). Similar equations have been used by Khan et al. (2004) andBardenhagen et al. (1997) previously. The modification was performed to predict theactual viscosity in the material for a specific strain rate as well as the operating tempera-ture of the polymer during the experiment. The modified viscosity function g, is given inthe following form:

gð_e; T Þ ¼ g1 þ g0 � g1

1þ ðaj_ejÞ2 � j_ej105

� �d� �b

26664

37775

T r � T vf

T � T vf

� �m

; ð12Þ

where g0 is the viscosity at zero strain-rate and g1 is the asymptotic viscosity at _e ¼ 1. _e isthe strain rate (s�1) experienced by each dashpot (note that the strain rates in the twodashpots are not the same). The material parameters a, d and b serve to adjust the decayof shear thinning under increasing strain rate. The denominator is, however, never allowedto become negative. It is noted that the viscosity at zero strain-rate, potentially introducesphysical insight into the constitutive model through the molecular weight of the polymer(Painter and Coleman, 1997). The material parameter, m, adjusts the response of viscosityof the polymer with different temperatures. Here, Tvf is the Vogel–Fulcher temperatureterm, taken from the well-known Vogel–Fulcher-Tammann Law, which relates viscosityas a function of temperature. Tvf is usually taken 50 K below the experimental glass tran-sition temperature (�Tg � 50 K) for polymers (Rault, 2000). Tg is the glass transition tem-perature. Tr is the reference temperature or the room temperature in the present study. T isthe temperature at which the experiment is performed. In case of Adiprene-L100 the glasstransition temperature is 211 K (�80 �F). Overall, the temperature term used here is takenfrom the Khan–Huang–Liang equation (Khan et al., 2004), which has been successfullyused to predict the viscoplastic response of metal and alloys.

The behavior of the two springs E1 and E2, for the standard solid model during stressrelaxation, present a dependency on the strain, not on strain rate. Hence, it seems that thespring components of the modified standard solid model are composed of two distinctmodes, the loading and the relaxation mode. Based on this idea, the springs E1 and E2

in the modified standard solid model are redefined in the following form:

E1 ¼ c1 1þ T r � TT d

� �; ð13Þ

E2 ¼ c2jejn2 1þ T r � TT d

� �ð14Þ

Here, c1, c2, and n2 are material parameters and |e| is the true stain of the material. Tr, Td

and T are the room, the melting or decomposition (�473 K) and the operating tempera-tures, respectively. Note that higher refinement on the dependence on the strain could beobtained by splitting E1 in the same manner as E2. It should also be emphasized that manyother forms of nonlinearities, i.e., rubber elasticity, might be implemented instead of thesimple power law utilized in Eq. (14). However, this would lead to a more complex modeland correspondingly more number of material parameters.


Fig. 3 shows the ‘‘rheological’’ schematic of the modified standard solid model incorpo-rating the new concepts discussed above. The constitutive equation of the developed con-stitutive model, for room temperature, may finally be expressed as

c1g1

rþ _r ¼ c1 _eþc1g1

c2eðn2þ1Þ þ g2g1

c1 _eþ c2ðn2 þ 1Þen2 _e. ð15Þ

In summary, the one-dimensional constitutive relation given by Eq. (15) characterizesthe observed viscoelastic behavior of typical polymeric materials under finite deformation.The functional expressions, g1 and g2, are dependent on strain rate and temperature; theydecrease with increasing strain rate (shear thinning) and temperature (see Eq. (12)). Non-linearity characteristics of the stress relaxation process is incorporated via a nonlinearspring, this is represented by E2. Moreover, the constitutive equation (15) represents a flex-ible framework for future generalizations to three-dimensional modeling.

The stress response of the polymer for constant loading (constant strain rate) experi-ments was determined by solving the governing differential equation. This solution ofthe differential equation for constant loading is given by the equation below:

r ¼ ðg1 þ g2Þ_e 1� e�c1e=g1 _e� �

þ c2eðn2þ1Þ. ð16Þ

Also, based on the differential equation shown in (15), the stress drop during the roomtemperature relaxation is found out by substituting the strain rate equal to zero ð_e ¼ 0Þ.The stress in this case can be written as:

r ¼ c2eðn2þ1Þ0 þ r0 � c2e

ðn2þ1Þ0

� �e�c1t=g1 . ð17Þ

In this case the viscosity term given by Eq. (12) reduces to gð_e; T Þ ¼ g0 for experimentsat reference temperature.

4. Results and discussion

4.1. Application of the modified form of MSP to Adiprene-L100

Fig. 4 shows the relaxation response of Adiprene-L100 at different levels of strain. Thecorresponding true stress versus time plot can be seen in Fig. 5. The bilinear behavior inthe Ln–Ln space (see Fig. 6) exhibited by Adiprene-L100 during relaxation can be ade-quately captured in the stress–time space by the following expression

rðtÞ ¼ k1er1 þ k2tr2e; ð18Þwhere k1, r1, k2, and r2 are material parameters. Note that Eq. (18) does not perfectly fitthe original MSP form given by Eq. (8); instead of having a first term linear in the strain,the actual behavior of the polymer has a power law. The incorporation of this new featureresults into a more general, more flexible and more nonlinear constitutive relation. In es-sence, by assuming the stress–time relationship given by Eq. (18), the MSP constitutiveequation reduces to

rðtÞ ¼ k1er1ðtÞ þZ t

0

k2ðt � sÞr2 deðsÞds

ds. ð19Þ

The material parameters k1, r1, k2, and r2 for Adiprene-L100 were determined from thestress relaxation versus time data shown in Fig. 5. Table 1 shows the values of these

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

100

200

300

400

500

600

700

800

True Strain (m/m)

Tru

e (C

auch

y) S

tres

s (p

si)

Tru

e (C

auch

y) S

tres

s (M

Pa)

0.50.570.660.760.871 Stretch

0

1

2

3

4

5Strain Rate = 1 s-1

Fig. 4. True (Cauchy) stress–strain for relaxation experiments at different levels of strain at the engineering strainrate, or stretch rate, of 1 s�1 (compressive stresses and strains are assumed positive).

0 1000 2000 3000 4000 5000 6000 7000 80000

100

200

300

400

500

600

700

800

Time (seconds)

Tru

e (C

auch

y) S

tres

s (

psi

)

Tru

e (C

auch

y) S

tres

s (

MP

a)

0

1

2

3

4

5

59.1%

49.9%40.4%

28.7%

19.2%

9.3%

Fig. 5. True (Cauchy) stress–time for relaxation experiments at different levels of strain at the engineering strainrate, or stretch rate, of 1 s�1 (compressive stresses and strains are assumed positive).


-6 -4 -2 0 2 4 6 8 10 124.5

5

5.5

6

6.5

7

Ln [ Time (seconds) ]

Ln

[ T

rue

(Cau

chy)

Str

ess

(psi

) ]

Ln

[ T

rue

(Cau

chy)

Str

ess

(MP

a) ]

–0.48

0.02

0.52

1.02

1.52

2.02

59.1%49.9%40.4%

28.7%

19.2%

9.3%

Fig. 6. Ln–Ln plot of the relaxation stress and time depicting a bilinear response (compressive stresses and strainsare assumed positive).

Table 1Material constants for Adiprene-L100 using MSP

k1 272.31 psik2 814.29 psi/s�0.45

r1 0.365r2 �0.045


parameters. Fig. 7 shows the experimental data along with the corresponding correlationsfrom the modified form of MSP for all the stress relaxation experiments performed onAdiprene-L100. It is clear that the modified MSP constitutive model correlates very wellwith the stress relaxation response of Adiprene-L100 for small deformations; it is incapa-ble to model the relaxation response at large deformations.

Fig. 8 shows the experimental true (Cauchy) stress versus true strain for constant strainrate loading on Adiprene-L100 at engineering strain rates (or stretch rates) of_eeð _kÞ ¼ 10�5; 10�4; 10�2; 100; and 5000 s�1 along with the predictions from the MSPrelation. It is clear from Fig. 8 that the MSP constitutive relation provides reasonable pre-dictions for quasi-static strain rate deformations. However, it is unable to predict theactual behavior of the polymer Adiprene-L100 for deformations under dynamic strain rateregime.

During these experiments the engineering strain rate or stretch rate was held constant.Since, the true strain rate changes with deformation in the specimen, in model predictionsas well as the viscosity functions it has been considered piecewise-constant for each 10% ofdeformation. The values of true strain rate at the two ends of each piecewise-constantintervals is averaged and used for that period.

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

100

200

300

400

500

600

700

800

Time (seconds)

Tru

e (C

auch

y) S

tres

s (p

si)

Tru

e (C

auch

y) S

tres

s (M

Pa)

0

1

2

3

4

5

59.1%

49.9%

40.4%

28.7%

19.2%

9.3%

Line: Experimental DataSymbols: Correlations

Fig. 7. MSP correlations during relaxation at different levels of strain (compressive stresses and strains areassumed positive).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

200

400

600

800

1000

1200

1400

1600

1800

2000

True Strain (m/m)

Tru

e (C

auch

y) S

tres

s (p

si)

Tru

e (C

auch

y) S

tres

s (M

Pa)

0.50.570.660.760.871Stretch

0

2

4

6

8

10

12

5000 s-1

1 s-1

0.01 s-1

0.0001 s-1

0.00001 s-1


Fig. 8. MSP correlations during constant strain rate loading experiments (compressive stresses and strains areassumed positive). Strain-rates shown are corresponding engineering rates, or stretch rates.



An observation should be made before continuing with the approach and application ofthe differential model. In the preceding study of the modified form of MSP, the materialconstants were determined from stress relaxation data. During a stress relaxation experi-ment no changes in strain occur with time. On the other hand, during a constant strain orstress rate loading the strain does change with time and the rate of change of straindepends on the rate of loading. It could be argued that there are significant effects thatare caused by changes of strain with time. Given that these effects do not arise during astress relaxation response, models that use material parameters determined from this kindof experiments might overlook certain features of the material. When applicable, a possi-ble improvement might be reached by using creep data, instead of stress relaxation data, todetermine the material parameters of these models as shown by other researchers (Smartand Williams, 1972).

4.2. Application of the modified standard solid model to Adiprene-L100

The stress relaxation data (Fig. 5) were used to calculate different material parametersinvolved in Eq. (17) such as c1, c2, n2 and g0. r0 is the initial stress level at which the relax-ation begins. Also, e0 represents the fixed strain at which the relaxation is performed. Leastsquare optimization techniques were used to get these material parameters. Using Eqs.(13) and (14), with the temperature term as unity at room temperature the linear andnon-linear spring constants were found out to be

E1 ¼ 1143 psi ð20ÞE2ðeÞ ¼ 630jej�0:385 psi. ð21Þ

The viscosity function for the first dashpot at zero strain rate was found to beg1(0) = g(0)1 = 155,263 psi s.

The model correlations obtained for the modified standard solid model along with theexperimental data during relaxation are shown in Fig. 9. Note that the asymptotic behav-ior of the model presents a good correlation with the experimental data. The model cap-tures even the transient part of the stress relaxation well. Although the results observed arequite close to observations still some scope for improvement is possible. In order to cap-ture more accurately the relaxation response of Adiprene-L100 with the modified standardsolid model, the addition of more relaxation times would be required. This could beaccomplished by the addition of more Maxwell elements.

The remaining material parameters were determined from uniaxial compression exper-iments under quasi-static and dynamic loadings at constant strain rates. However, due tothe nonlinearity introduced by Eq. (21), there is not a closed form solution for the constantstrain rate deformation of the modified standard solid model. This, in turn, implies theimpossibility of determining the material parameters g1, g0, a, d and b analytically.Numerical methods need to be employed in order to obtain these material constants.The approach followed in this study was to apply a least square-fit type method to theexperimental data on Adiprene-L100 under uniaxial compression at engineering strain rate(stretch rate) loadings of _eeð _kÞ ¼ 10�5; 10�4; 10�2; 100 and 5000 s�1. The results yieldedthe following material constants for the two dashpots. Note that the equation is validfor room temperature experiments only and since the reference temperature for modelingis taken as room temperature, the temperature term in the viscosity equation becomesunity

0 1000 2000 3000 4000 5000 6000 7000 80000

100

200

300

400

500

600

700

800

900

1000

Time (seconds)

Tru

e (C

auch

y) S

tres

s (p

si)

Tru

e (C

auch

y) S

tres

s (M

Pa)

0

1

2

3

4

5

6

59.1%49.9%40.4%

28.7%

19.2%

9.3%


Fig. 9. True (Cauchy) stress–time for relaxation experiments with modified standard model correlations atdifferent levels of strain at the engineering strain rate, or stretch rate, of 1 s�1 (compressive stresses and strains areassumed positive).


g1 ¼ 3:42þ 155263 � 3:42

1þ ð3790� j_ed1jÞ2 � j_ed1j105

� �2� �0:414

26664

37775; ð22Þ

g2 ¼ 10:73þ 151160� 10:73

1þ ð1411� j_ejÞ2 � _ej j105

� �2� �0:907

26664

37775. ð23Þ

The strain rate in dashpot 1 ð_ed1Þ was determined iteratively by finding the stress in thatparticular arm (since the strain rate in that dashpot is the difference of the total strain rateð_eÞ minus the strain rate experienced by the spring E1). This strain rate was then utilized tofind the new set of constants for the dashpot with the corrected strain rate. The strain ratein the spring, E1, for the quasi-static experimental results was found to be almost negligi-ble. However, for dynamic experiment it was found to be comparable to the strain rate inthe dashpot. Fig. 10 shows the experimental true (Cauchy) stress versus true strain for theengineering strain rate (stretch rate) loading on Adiprene-L100 at strain rates of_eeð _kÞ ¼ 10�5; 10�4; 10�2; 100 and 5000 s�1 along with the correlations from the modifiedstandard solid model. The behavior exhibited by the model is in reasonable agreementwith the observed response for all quasi-static, as well as for the dynamic strain rate exper-iments. This implies that the dependency on the strain rate extracted from the quasi-staticand dynamic strain rate loading experiments, i.e., the modified term, g, remains true in allthe regimes.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

500

1000

1500

2000

True Strain (m/m)

Tru

e (C

auch

y) S

tres

s (p

si)

Tru

e (C

auch

y) S

tres

s (M

Pa)

0.50.570.660.760.871

Stretch

0

2

4

6

8

10

12

5000 s-1

1 s-1

0.01 s-1

0.0001 s-1

0.00001 s-1


Fig. 10. True (Cauchy) stress–strain for uniaxial loadings with modified standard model correlations at differentstrain rates (compressive stresses and strains are assumed positive). Strain-rates shown are correspondingengineering rates, or stretch rates.


As mentioned before, the viscosity terms as well as the spring moduli were modified toinclude the effect of temperature on the response of the polymer (Eqs. (12)–(14)). Thismodification enabled modeling of the thermo-mechanical response of the polymer forlarge strains. While modeling the response of the polymer with different operating temper-atures, the constants achieved during room temperature experiment correlations were keptconstant at all times. The temperature sensitivity parameters, m, of the viscosity equationsin both the dashpots were allowed to adjust themselves so as to produce good agreementwith the experimental data. This was performed using a curve fitting scheme and leastsquare optimization. The temperature response of the polymer along with the model cor-relations can be found in Fig. 11. The model was able to correlate well with the observedresponse even for the experiment performed above the glass transition temperature. Theadded temperature terms in the calculation of viscosity in the dashpots as well as the linearand nonlinear springs were able to capture the response reasonably well. All material con-stants for the newly developed viscoelastic model are shown in Table 2.

In order to further validate the capability of the modified standard solid model in pro-ducing good agreement of the observed response, predictions were performed on a com-pressive strain rate jump experiment on Adiprene-L100. In this experiment at roomtemperature, the engineering strain rate (stretch rate) loading of _eeð _kÞ ¼ 10�4 s�1 wasmaintained constant up to a strain level of e = 24%, followed by a jump in the strain rate(stretch rate) to _eeð _kÞ ¼ 10�2 s�1 and then maintained constant to a strain level of e = 45%.Constants determined utilizing the prior experiments were not changed to predict theresponse of the material under the given conditions. Fig. 12 illustrates the observed true(Cauchy) stress versus true strain response for the jump experiment along with the model

0 0.1 0.2 0.3 0.4 0.5 0.60

500

1000

1500

2000

2500

True Strain(m/m)

Tru

e (C

auch

y) S

tres

s (p

si)

Tru

e (C

auch

y) S

tres

s (M

Pa)

0

2

4

6

8

10

12

14

16

0.550.620.700.770.891Stretch

199K

266 K

294 K358 K405 K

Line : Experimental DataSymbols: Correlations

Fig. 11. Uniaxial experiments performed at 0.1 s�1 engineering strain rate or stretch rate and at differenttemperatures along with the modified standard model correlations (compressive stresses and strains are assumedpositive).

Table 2Material constants for Adiprene-L100 using MSSM

c1 c2 n2 g(inf)1 g(0)1 a1 d1 b1 m1

1143 psi 630 psi �0.385 3.42 psi s 155,263 psi s 3790 2 0.414 1.84g(inf)2 g(0)2 a2 d2 b2 m2

10.73 psi s 151,160 psi s 1411 2 0.907 5.48


predictions. The agreement between the two was found to be reasonably good. Furthervalidation of the capability of the differential model was carried out for a multiple stepstress relaxation experiment, which was performed at room temperature. Here again,the constants determined using the earlier set of experiments were not changed to predictthe response of the material under constant rate of loading or during relaxation periods.Fig. 13 shows the experimental stress–strain relationship for a constant engineering strainrate(stretch rate) loading of _eeð _kÞ ¼ 100 s�1 with multiple step 2 h stress relaxation pro-cesses around e = 15%, 30%, 45% and 60% strain levels along with the corresponding pre-dictions from the modified standard solid model. Again, the agreement between theexperimental observations and the predictions from the model are very good.

Free end pure torsion experiments were performed on the Adiprene-L100 samples atdifferent shear strain rates (see Fig. 14). The axial force was kept zero during the lengthof the experiment. The shear stress and strain values were calculated from the experimen-tal data. The new modified viscoelastic model was used to calculate the effective von-Mises

0 0.1 0.2 0.3 0.4 0.50

100

200

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500

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700

800

True Strain (m/m)

Tru

e (C

auch

y) S

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s (p

si)

Tru

e (C

auch

y) S

tres

s (M

Pa)

0.610.670.740.820.911Stretch

0

1

2

3

4

5Line: Experimental DataSymbols: Predictions

Jump

0.0001 s-1

0.01 s-1

Fig. 12. True (Cauchy) stress–strain for strain rate jump (0.0001–0.01 s�1) with modified standard solid modelpredictions (compressive stresses and strains are assumed positive). Strain-rates shown are correspondingengineering rates or stretch rates.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

100

200

300

400

500

600

700

800

True Strain (m/m)

Tru

e (C

auch

y) S

tres

s (p

si)

Tru

e (C

auch

y) S

tres

s (M

pa)

0.50.570.660.760.871

Stretch

0

1

2

3

4

5

2 HrRelaxation

Line: Experimental DataSymbols: Predictions

Fig. 13. Multiple step relaxation experiment at 1 s�1 engineering strain rate or stretch rate with 2 h relaxationperiod along with the modified standard model predictions (compressive stresses and strains are assumedpositive).


0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

50

100

150

200

250

300

350

400

Shear Strain

Sh

ear

Str

ess

(psi

)

Sh

ear

Str

ess

(Mp

a)

0

0.5

1

1.5

2

2.5

γ =100 s-1

γ=10-2 s-1

γ=10-4 s-1.

.

.

Symbols: Experimental DataLine: Predictions

Fig. 14. Free end torsion experiments at various shear strain rates at room temperature along with modelpredictions (note that the material constants determined using previous experiments were not changed).


equivalent stress and strain at the given shear strain rates and compared with the experi-mental observations. In these predictions from the model, the constants were not changedfrom the ones determined during the room temperature compression experiments. Themodel predictions are close to the observed response. The use of von-Mises equivalentstress is an assumption which needs to be investigated later.

5. Conclusions

The compressive uniaxial behavior of the polymer Adiprene-L100 was modeled forquasi-static and dynamic operating conditions, as well as, the stress relaxation experimentsusing a modified form of MSP and a modified standard solid model developed in thisstudy. Modified standard solid model was also used to capture the response of Adi-prene-L100 at different temperatures under quasi-static loadings. The correlationsobtained for the modified standard solid model were much better than those by the mod-ified MSP especially in the dynamic strain rate regime and/or at finite deformations. Forthe stress drop during relaxation periods, the two models were found to be quite compa-rable for lower strain levels, but at large deformations the new model had much betterresults. Modeling of experimental observations at a specific strain rate for different temper-atures was also performed on the polymer by employing springs with temperature depen-dent constants and dashpots with viscosity term to incorporate temperature effects. Thenew model correlations were in fairly good agreement with the experimental observations.Predictions, using the model constants determined from the uniaxial compression experi-ments, were performed on a compressive strain rate jump experiment, on compressivemultiple relaxation experiment and also on free end pure torsion experiments. These pre-dictions, in each case, were very close to the experimental observations. Thus, the modelwas able to predict the response of the polymeric material very well for quasi-static


experiments at room as well as above and below the glass transition temperature regimes,over a wide range of strain rates, and single and multiple stress relaxation experiments.

This modified viscoelastic new model has a foundation on the infinitesimal linear theory(standard solid model) with a semi-empirical modification based on experimental observa-tions. This implies that the various parameters used in the model have a clear physicalinsight. In synthesis, the new model developed in this work presents a simple and flexibleframework to model nonlinear viscoelastic systems under finite deformation over a widerange of strain rates and temperatures. This model can also be a basis for later generaliza-tions to three-dimensional cases.

References

Argon, A.S., Butalov, V.V., Mott, P.H., Suter, U.W., 1995. Plastic deformation in glassy polymers by atomisticand microscopic simulations. J. Rheol. 39, 377–399.

Bardenhagen, S.G., Stout, M.G., Gray, G.T., 1997. Three-dimensional, finite deformation, viscoplasticconstitutive models for polymeric materials. Mech. Mater. 25, 235–253.

Bergstrom, J.S., Boyce, M.C., 1998. Constitutive modeling of the large strain time-dependent behavior ofelastomers. J. Mech. Phys. Solids 46, 931–954.

Bird, R.B., Armstrong, R.C., Hassager, O., 1977Dynamics of Polymeric Liquids, vol. 1. Wiley, New York.Boyce, M.C., Parks, D.M., Argon, A.S., 1988. Large inelastic deformation of glassy polymers, Part I: rate

dependent constitutive model. Mech. Mater. 7, 15–33.Caruthers, J.M., Adolf, D.B., Chambers, R.S., Shrikhande, P., 2004. A thermodynamically consistent, nonlinear

viscoelastic approach for modeling glassy polymers. Polymer 45, 4577–4597.Christensen, R.M., 1982. Theory of Viscoelasticty: An Introduction. Academic Press, New York.Colak, O., 2005. Modeling deformation behavior of polymers with viscoplasticity theory based on overstress. Int.

J. Plasticity 21, 145–160.Drozdov, A.D., Dorfmann, A., 2003. A micro-mechanical model for the response of filled elastomers at finite

strains. Int. J. Plasticity 19, 1037–1067.Ehlers, W., Markert, B., 2003. A macroscopic finite strain model for cellular polymers. Int. J. Plasticity 19, 961–

976.Findley, W.N., 1968. Product form of kernel functions for nonlinear viscoelasticity of PVC under constant rate

stressing. Trans. Soc. Rheol. 12 (2), 217–242.Flugge, W., 1967. Viscoelasticity. Blaisdel Publishing Company.Gray III, G.T., Blumenthal, W.R., Trujillo, C.P., Carpenter, R.W., 1997. Influence of temperature and strain rate

on the mechanical behavior of Adiprene-L100. Report No. LA-UR-97-0905. Las Alamos NationalLaboratory.

Green, A.E., Rivlin, R.S., 1957. The mechanics of non-linear materials with memory; Part I. Arch. Ration. Mech.Anal. 1, 1.

Haupt, P., Lion, A., Backhaus, E., 2000. On the dynamic behavior of polymers under finite strains: constitutivemodeling and identification of parameters. Int. J. Solids Struct. 37, 3633–3646.

Khan, A.S., Suh, Y.S., Kazmi, R., 2004. Quasi-static and dynamic loading responses and constitutive modeling oftitanium alloys. Int. J. Plasticity 20, 2233–2248.

Khan, A.S., Zhang, H., 2001. Finite deformation of a polymer and constitutive modeling. Int. J. Plasticity 17,1167–1188.

Krempl, E., Khan, F., 2003. Rate (time)-dependent deformation behavior: an overview of some properties ofmetals and solid polymers. Int. J. Plasticity 19, 1069–1095.

Lai, J.S.Y., Findley, W.N., 1968. Stress relaxation of nonlinear viscoelastic material under uniaxial strain. Trans.Soc. Rheol. 12, 259–280.

Liang, R., Khan, A.S., 1999. A critical review of experimental results and constitutive models for BCC and FCCmetals over a wide range of strain rates and temperatures. Int. J. Plasticity 15 (9), 963–980.

Lion, A., 2001. Thermomechanically consistent formulations of the standard linear solid using fractionalderivatives. Arch. Mech., 53.

Lion, A., Kardelky, C., 2004. The Payne effect in finite viscoelasticity: constitutive modeling based on fractionalderivatives and intrinsic time scales. Int. J. Plasticity 20, 1313–1345.


Liu, Y., Gall, K., Dunn, M.L., Greenberg, A.R., Diani, J., 2006. Thermomechanics of shape memory polymers:uniaxial experiments and constitutive modeling. Int. J. Plasticity 22 (2), 279–313.

Lockett, F.J., 1972. Nonlinear Viscoelastic Solids. Academic Press, London.Lopez-Pamies, O., Khan, A.S., 2002. Time and temperature dependent response and relaxation of a soft polymer.

Int. J. Plasticity 18, 1359–1372.Lubarda, V.A., Benson, D.J., Meyers, M.A., 2003. Strain-rate effects in rheological models of inelastic response.

Int. J. Plasticity 19, 1097–1118.Makradi, A., Ahzi, S., Gregory, R.V., Edie, D.D., 2005. A two-phase self-consistent model for the deformation

and phase transformation behavior of polymers above the glass transition temperature: application to PET.Int. J. Plasticity 21, 741–758.

McGuirt, C., Lianis, G., 1970. Constitutive equations for viscoelastic solids under finite uniaxial and biaxialdeformations. Trans. Soc. Rheol. 14 (2), 117–134.

Ogden, R.W., 1997. Non-linear Elastic Deformations. Dover Publications Inc., Mineola, New York.Painter, P.C., Coleman, M.M., 1997. Fundamentals of Polymer Science: An Introductory Text, second ed.

Technomic Publishing Co., Inc..Pipkin, A.C., Rogers, T.G., 1968. A nonlinear integral representation for viscoelastic behavior. J. Mech. Phys.

Solids 16, 59–72.Rault, J., 2000. Origin of the Vogel–Fulcher–Tammann law in glass-forming materials: the a–b bifurcation. J.

Non-Cryst. Solids 271, 177–217.Reese, S., 2003. A micromechanically motivated material model for the thermo-viscoelastic material behavior of

rubber-like polymers. Int. J. Plasticity 19, 909–940.Smart, J., Williams, J.G., 1972. A comparison of single-integral non-linear viscoelasticity theories. J. Mech. Phys.

Solids 20, 313–324.Ward, I.M., Sweeney, J., 2004. An Introduction to the Mechanical Properties of Solid Polymers, second ed.

Wiley, New York.

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